"I think the court would be likely to award something here."
I'm not so sure about that.
I read the case in the article that you linked to (in which the claimants were unsuccessful anyway).
In most cases, this situation arises when two parties are still negotiating about a contract and they say "OK, you start work now even though we don't have a contract and we'll finalise the details and the costs at a later point". In that case if one party does the work and for some reason the two parties cannot agree on a contract then they will still get paid for doing the work.
When you talk about unjust enrichment it was claimed in the case linked to that the Defendant had approached the Claimant asking for advice. But there is no general right to payment for requested services in the absence of a contract.
However, if a person requests a service knowing that it is only done in return for money then they will have to pay.
In contrast, if somebody takes a risk and offers to do some work in the hope of getting a contract from it then they do not need to be paid.
In the case that was linked, the judge decided that the Claimant had done the work in the hope that they would later get a contract.
Moorgate Capital (Corporate Finance) Ltd v H.I.G. European Capital Partners LLP [2019] EWHC 1421 (Comm)
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But, to bring it all back to the OP's situation:
Imagine if the gardener had come round and OP agreed for him to cut the lawn in the front garden for £X. While he was there, OP and the gardener start discussing how much he would charge for cutting the rear lawn.
The gardener then says "While we're discussing the price, shall I just start mowing the rear lawn anyway?" to which the OP agrees.
They then cannot agree a price but the gardener has already started mowing the rear lawn. In this situation, the gardener can claim a reasonable payment for the extra work done.
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Or another situation, if the gardener comes up to OP and says "Hello, I run a gardening business where I charge people to mow their lawns. Would you like me to mow your lawn?"
OP says "Yes please" without enquiring about the price. She's on the hook for paying in that situation.
In contrast, if the gardener had said "Hello, I run a lawn mowing service. Would you like a demonstration of how good I am at mowing lawns?"
OP says "Yes, please" and the gardener mows the front lawn. OP then says "Well, that wasn't good enough for me so I'm not going to hire you". In that situation the gardener cannot claim for the work he does in showing a potential customer his capability or skills.
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However, the actual situation with the OP, where the gardener did not make any contact at all with her before doing the work is just so far away from those situations that there is no way that the gardener would be able to claim for payment.